College Algebra (6th Edition)

Published by Pearson
ISBN 10: 0-32178-228-3
ISBN 13: 978-0-32178-228-1

Chapter 4 - Exponential and Logarithmic Functions - Exercise Set 4.3: 32

Answer

$\frac{1}{5}log_{10}x-\frac{1}{5}log_{10}y$

Work Step by Step

We know that $log(x)$ is a common logarithm with an understand base of 10. Therefore, $log\sqrt[5] \frac{x}{y}=log_{10}\sqrt[5] \frac{x}{y}=log_{10}\frac{\sqrt[5] x}{\sqrt[5] y}$. Based on the quotient rule of logarithms, we know that $log_{b}(\frac{M}{N})=log_{b}M-log_{b}N$ (where $b$, $M$, and $N$ are positive real numbers and $b\ne1$). Therefore, $log_{10}\frac{\sqrt[5] x}{\sqrt[5] y}=log_{10}\sqrt[5] x-log_{10}\sqrt[5] y=log_{10}x^{\frac{1}{5}}-log_{10}y^{\frac{1}{5}}$. According to the power rule of logarithms, we know that $log_{b}M^{p}=plog_{b}M$ (when $b$ and $M$ are positive real numbers, $b\ne1$, and $p$ is any real number). Therefore, $log_{10}x^{\frac{1}{5}}-log_{10}y^{\frac{1}{5}}=\frac{1}{5}log_{10}x-\frac{1}{5}log_{10}y$.
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